How do I change the angle of the axis without changing the angle of the object?
Blender's precision modelling tools are a bit lacking unfortunately but you can do this with an auxiliary Empty object.

Create an empty object, align it with an one of the vertex of the object you wish to correct the alignment.

Rotate it and use snapping to align it with the desired edge.

Now copy the empty object's rotation to the other object, either by rotating it in object mode that same number of degrees and applying it, or entering edit mode and rotating all geometry that same amount.